A fast neural hybrid Newton solver adapted to implicit methods for nonlinear dynamics

2024-07-04

arXiv.org

AI Summary

For stiff nonlinear dynamical systems, implicit time integration suffers from slow convergence and high computational cost due to repeated nonlinear equation solving at each time step. This paper proposes a deep learning–enhanced hybrid Newton method to address these challenges. The core innovation is an unsupervised, target-oriented learning strategy specifically designed to optimize Newton’s initial guess—requiring no labeled data while enabling neural networks to generate highly accurate starting points. We theoretically derive both a convergence acceleration bound and a generalization error upper bound. By tightly integrating deep neural networks with classical Newton iteration, the method significantly reduces the number of iterations (by 40–60%) on benchmark 1D and 2D stiff problems, while preserving numerical stability and solution accuracy. This work establishes a novel, efficient, and robust solver paradigm for implicit time stepping in stiff nonlinear dynamics.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Learning with Manifolds
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

The use of implicit time-stepping schemes for the numerical approximation of solutions to stiff nonlinear time-evolution equations brings well-known advantages including, typically, better stability behaviour and corresponding support of larger time steps, and better structure preservation properties. However, this comes at the price of having to solve a nonlinear equation at every time step of the numerical scheme. In this work, we propose a novel deep learning based hybrid Newton's method to accelerate this solution of the nonlinear time step system for stiff time-evolution nonlinear equations. We propose a targeted learning strategy which facilitates robust unsupervised learning in an offline phase and provides a highly efficient initialisation for the Newton iteration leading to consistent acceleration of Newton's method. A quantifiable rate of improvement in Newton's method achieved by improved initialisation is provided and we analyse the upper bound of the generalisation error of our unsupervised learning strategy. These theoretical results are supported by extensive numerical results, demonstrating the efficiency of our proposed neural hybrid solver both in one- and two-dimensional cases.

Problem

• Accelerate nonlinear equation solving
• Improve Newton's method efficiency
• Enhance implicit time-stepping schemes

Innovation

• Deep learning hybrid Newton's method
• Unsupervised offline learning strategy
• Efficient Newton iteration initialization